A Bernoulli Process is a sequence of iid (independent, identically-distributed) random variables x1,x2,...,xi,..., where the probability mass function of xi is
xi = S(success) with probability p,
xi = F(failure) with probability q,
with p+q = 1.
Depending on the application, (S,F) may be (1,0), (H,T), (true, false), etc.
Here are two obvious ways to generate a (1,0) Bernoulli process in Matlab:
- Vectorized Form: bernp1 = rand(n,1) <= p; Process vector bernp1
- Component Form: for k = 1:n, bernp2 = rand <= p; Process scalar bernp2; end;
The "best" answer will depend on the application and computer at hand.
If the computer has lots of memory then the vectorized version is the simplest, and may be the fastest.
If "Process vector or scalar" takes a lot of time then the time to generate bernp by either method may be negligible.
If a fixed amount of memory is available for bernp then we must use the component form or a mini-vectorized form.
Using rand to generate 1 random bit seems wasteful, given that rand returns a double precision floating point number which has about 53 random bits. Can we use rand more efficiently?
My interest in the Bernoulli process was sparked by re-reading Feller Volume 1. At the bottom of page 18 Feller says: "In this volume we shall consider only discrete sample spaces". A closer examination shows that Feller derives (nearly) everything in this book from the Bernoulli process.
Derek O'Connor
